New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions

This paper investigates the forced Duffing equation with integral boundary conditions. Its approximate solution is developed by combining the homotopy perturbation method (HPM) and the reproducing kernel Hilbert space method (RKHSM). HPM is based on the use of the traditional perturbation method and the homotopy technique. The HPM can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully linear boundary value problems. Therefore, the forced Duffing equation with integral boundary conditions can be solved using advantages of these two methods. Two numerical examples are presented to illustrate the strength of the method.     

Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems

In this paper, based on homotopy perturbation method (HPM) and reproducing kernel method (RKM), a new method is presented for solving nonlinear systems of second order boundary value problems (BVPs). HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKM is also an analytical technique, which can solve powerfully linear BVPs. Homotopy perturbation-reproducing kernel method (HP-RKM) combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear BVPs. Three numerical examples are presented to illustrate the strength of the method.     

On the numerical solution of singular two‐point boundary value problems: A domain decomposition homotopy perturbation approach

This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two‐point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate.     

Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique

In this work, we implement a relatively analytical technique, the homotopy perturbation method (HPM), for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. He’s homotopy perturbation method (HPM) which does not need small parameter is implemented for solving the differential equations. It is predicted that HPM can be found widely applicable in engineering.     

Analytical solution of time‐fractional Navier–Stokes equation in polar coordinate by homotopy perturbation method

In this letter, we implement a relatively new analytical technique, the homotopy perturbation method (HPM), for solving linear partial differential equations of fractional order arising in fluid mechanics. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. Some numerical examples are presented to illustrate the efficiency and reliability of HPM. He's HPM, which does not need small parameter is implemented for solving the differential equations. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants that can be determined by imposing the boundary and initial conditions. It is predicted that HPM can be found widely applicable in engineering. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Some issues on HPM and HAM methods: A convergence scheme

The homotopy method for the solution of nonlinear equations is revisited in the present study. An analytic method is proposed for determining the valid region of convergence of control parameter of the homotopy series, as an alternative to the classical way of adjusting the region through graphical analysis. Illustrative examples are presented to exhibit a vivid comparison between the homotopy perturbation method (HPM) and the homotopy analysis method (HAM). For special choices of the initial guesses it is shown that the convergence-control parameter does not cover the HPM. In such cases, blindly using the HPM yields a non convergence series to the sought solution. In addition to this, HPM is shown not always to generate a continuous family of solutions in terms of the homotopy parameter. By the convergence-control parameter this can however be prevented to occur in the HAM.     

Approximate explicit solution of Camassa-Holm equation by He’s homotopy perturbation method

In this paper, the homotopy perturbation method (HPM) is employed to solve Camassa-Holm equation. Approximate explicit solution is obtained. Comparing the approximate solution with its exact solution shows the applicability, accuracy and efficiency of HPM in solving nonlinear differential equations. It is predicted that HPM can be widely applied in applied mathematics and engineering problems.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

Spectral homotopy analysis method and its convergence for solving a class of nonlinear optimal control problems

A combination of the hybrid spectral collocation technique and the homotopy analysis method is used to construct an iteration algorithm for solving a class of nonlinear optimal control problems (NOCPs). In fact, the nonlinear two-point boundary value problem (TPBVP), derived from the Pontryagin’s Maximum Principle (PMP), is solved by spectral homotopy analysis method (SHAM). For the first time, we present here a convergence proof for SHAM. We treat in detail Legendre collocation and Chebyshev collocation. It is indicated that Legendre collocation gives the same numerical results with Chebyshev collocation. Comparisons are made between SHAM, Matlab bvp4c generated results and results from literature such as homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM) and differential transformations.     

Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations

In this paper, homotopy perturbation method (HPM) and variational iteration method (VIM) are applied to solve nonlinear oscillator differential equations. Illustrative examples reveal that these methods are very effective and convenient for solving nonlinear differential equations. Moreover, the methods do not require linearization or small perturbation. Comparisons are also made between the exact solutions and the results of the homotopy perturbation method and variational iteration method in order to prove the precision of the results obtained from both methods mentioned.     

The travelling wave solutions for non-linear initial-value problems using the homotopy perturbation method

In this article, we have used the homotopy perturbation method (HPM) to find the travelling wave solutions for some non-linear initial-value problems in the mathematical physics. These problems consist of the Burgers–Fisher equation, the Kuramoto–Sivashinsky equation, the coupled Schordinger KdV equations and the long–short wave resonance equations together with initial conditions. The results of these problems reveal that the HPM is very powerful, effective, convenient and quite accurate to the systems of non-linear equations. It is predicted that this method can be found widely applicable in engineering and physics.     

New Analytic Solution of Von Karman Swirling Viscous Flow

In this paper, an analytic method for highly nonlinear problems, namely, homotopy perturbation (HPM) method, is used to solve the Von Karman swirling viscous flow, governed by a set of two fully coupled differential equations with strong nonlinearity. An explicit, purely analytic and uniformly valid solution is given. Comparison of the results obtained by the method reveals that HPM is more effective and easy to use.     

Application of He's methods for laminar flow in a porous‐saturated pipe

In this paper the problem of fully developed laminar steady forced convection inside a porous‐saturated pipe with uniform wall temperature is presented and the homotopy perturbation method (HPM) and the variational iteration method (VIM) are employed to solve the differential equations governing the problem. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and science. Copyright © 2009 John Wiley & Sons, Ltd.     

Approximate analytical solutions of reaction–diffusion equations with exponential source term: Homotopy perturbation method (HPM)

In this letter, the solutions of some nonlinear differential equations have been obtained by means of the homotopy perturbation method (HPM). Applications of the homotopy method to some nonlinear reaction–diffusion equations with exponential source term show rapid convergence of the sequence constructed by this method to the exact solutions.     

Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method

In this work, the homotopy analysis method (HAM), one of the most effective method, is implemented for finding approximate solutions of the Burger and regularized long wave (RLW) equations. Comparisons are made between the results of the proposed method and homotopy perturbation method (HPM). It illustrates the validity and the great potential of the homotopy analysis method in solving nonlinear partial differential equations.     

Laplace transform for the solution of higher order deformation equations arising in the homotopy analysis method

A new analytic approach for solving nonlinear ordinary differential equations with initial conditions is proposed. First, the homotopy analysis method is used to transform a nonlinear differential equation into a system of linear differential equations; then, the Laplace transform method is applied to solve the resulting linear initial value problems; finally, the solutions to the linear initial value problems are employed to form a convergent series solution to the given problem. The main advantage of the new approach is that it provides an effective way to solve the higher order deformation equations arising in the homotopy analysis method.     

Homotopy analysis method for singular IVPs of Emden–Fowler type

In this paper, approximate and/or exact analytical solutions of singular initial value problems (IVPs) of the Emden–Fowler type in the second-order ordinary differential equations (ODEs) are obtained by the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. It is shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.     

Numerical study of homotopy‐perturbation method applied to Burgers equation in fluid

In this article, the problem of Burgers equation is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. Comparison is made between the HPM and Exact solutions. The obtained solutions, in comparison with the exact solutions, admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

An Approximation of the Analytical Solution of the Linear and Nonlinear Integro-Differential Equations by Homotopy Perturbation Method

This paper aims to introduce an analytic technique, namely the Homotopy perturbation method (HPM) for the solution of integro-differential equations. From the computational viewpoint, the comparison shows that the homotopy perturbation method is efficient and easy to use.